Let $x_0$ be an unbounded self-adjoint operator such that the Brown measure of $x_0$ exists in the sense of Haagerup and Schultz. Also let $\tildeσ_α$ and $σ_β$ be semicircular variables with variances $α\geq 0$ and $β>0$ respectively. Suppose $x_0$, $σ_α$, and $\tildeσ_β$ are all freely independent. We compute the Brown measure of $x_0+\tildeσ_α+iσ_β$, extending the recent work which assume $x_0$ is a bounded self-adjoint random variable. We use the PDE method introduced by Driver, Hall and Kemp to compute the Brown measure. The computation of the PDE relies on a charaterization of the class of operators where the Brown measure exists. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+σ_{α+β}$. We also compute the example where $x_0$ is Cauchy-distributed.